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Name:

Lab Section:

MATH 215 – Fall 2004

FINAL EXAM

Show your work in this booklet.

Do NOT submit loose sheets of paper–They won’t be graded

Problem Points Score

1 15 2 10 3 25 4 10 5 15 6 15 7 10

TOTAL 100

Some useful trigonometric identities:

sin2 θ + cos2 θ = 1 cos 2θ = cos2 θ − sin2 θ sin 2θ = 2 sin θ cos θ

sin2 θ = 1− cos 2θ

2 cos2 θ =

1 + cos 2θ

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Spherical coordinates:

x = ρ cos(θ) sin(φ) y = ρ sin(θ) sin(φ) z = ρ cos(φ)

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Problem 1. (15 points) This problem is about the function

f(x, y, z) = 3zy + 4x cos(z).

(a) What is the rate of change of the function of f at (1, 1, 0) in the direction from this point to the origin?

(b) Give an approximate value of f(0.9, 1.2, 0.11).

CONTINUED ON THE NEXT PAGE 2

(c) Recall that f(x, y, z) = 3zy + 4x cos(z). The equation f(x, y, z) = 4 implicitly defines z as a function of (x, y), if we agree that z = 0 if

(x, y) = (1, 1). Find the numerical values of the derivatives

∂z

∂x (1, 1) and

∂z

∂y (1, 1).

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Problem 2. (10 points) (a) Find and classify the critical points of the function f(x, y) = −2x2 + 8xy − 9y2 + 4y − 4.

(b) Find the equation of the tangent plane to the surface z + 2x2 − 8xy + 9y2 − 4y = 0 at the point (2, 0,−8).

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Problem 3. (25 points) No partial credit. Evaluate each integral below. R > 0 and a > 0 are constants. Note: If you correctly understand the domains and/or the symmetries of the functions, many of the integrations become immediate. CIRCLE your answers.

(1) If D = [−1, 0]× [−1, 0] ∪ [0, 1]× [0, 1], then ∫∫

D

xy dA =

(2)

∫ a

0

∫ x

0

dy dx =

(3) If D is defined in polar coordinates by the inequalities: 0 ≤ r ≤ R, π/7 ≤ θ ≤ 8π/7,

then

∫∫

D

√

x2 + y2 dA =

(4)

∫ π/2

0

∫ π

0

∫ R

0

ρ2 sin(φ) dρ dθ dφ =

(5)

∫ π/4

0

∫ R

0

∫ a

0

r dz dr dθ =

5

Problem 4. (10 points) (a) Find the value of a such that the field on the plane

~F (x, y) = 〈axy + 1

x , x2〉

is conservative. Find a potential for the resulting field.

(b) Compute the line integral of the conservative field you found in part (a) over the curve which

is the image of 〈et 2

, t cos(2πt)〉, where 0 ≤ t ≤ 1.

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Problem 5. (9+6=15 points) (a) By a direct calculation, evaluate I =

∫

CR

dx+ x2ydy, where

CR is the triangle with vertices (0, 0), (0, R), (R, 0) oriented counterclockwise.

(b) Compute the value of I by evaluating a double integral, using Green’s theorem.

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Problem 6. (15 points) Let S be the portion of the surface x = 5 − y2 − z2 in the half space

x ≥ 1, oriented so that the normal vector at (5, 0, 0) is equal to ~ı. Let ~F (x, y, z) = 〈−1 , 1 , 0〉 (a constant vector field).

(a) Set up and evaluate an iterated double integral equal to ∫∫

S ~F · d~S.

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(b) It turns out that ~F = ∇× ~G, where ~G = 〈0 , z , −x〉. (You do not have to verify this.) Give an alternative calculation of the surface integral of part (a) by applying Stokes’ theorem.

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Problem 7. (10 points) Consider the plot of a vector field ~F = P (x, y)~ı+Q(x, y)~:

1

1.5

2

2.5

3

3.5

y

1 1.5 2 2.5 3 3.5

x

(a) Mark a point A at which curl( ~F ) > 0.

(b) Mark a point B at which div( ~F ) < 0.

(c) Sketch a closed curve C such that the circulation of ~F along C is positive.

(d) Mark two points S and T and two curves C1 and C2 from S to T such that the work done

by ~F in moving an object along those curves is positive for C1 and negative for C2.

(e) Can the field ~F be a gradient field? On the space below explain why or why not.

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